How much do you know? - Ratios, proportions, and percents - Data analysis

PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

How much do you know?
Ratios, proportions, and percents
Data analysis

LEARNING OBJECTIVES

After completing this chapter, you will be able to:

· Set up and solve a proportion for a missing value

· Use ratios to perform unit conversions

· Calculate percents and percent change

80/600 SmartPoints® (High Yield)

How much do you know?

Directions: Try the questions that follow. Show your work so that you can compare your solutions to the ones found in the Check Your Work section immediately after this question set. The “Category” heading in the explanation for each question gives the title of the lesson that covers how to solve it. If you answered the question(s) for a given lesson correctly, and if your scratchwork looks like ours, you may be able to move quickly through that lesson. If you answered incorrectly or used a different approach, you may want to take your time on that lesson.

1. image Seven out of every 250 students at a certain university who take a test are expected to score at least 90 percent. If the university gives this test to 12,000 students, how many would be expected to score at least 90 percent?

A. 176

B.224

C.300

D. 336

2. A homeowner wants to buy 81 square feet of grass for his yard, but the vendor he uses sells grass only by the square yard. How many square yards of grass does the homeowner need? (1 yard = 3 feet)

A. 9

B.27

C.243

D. 729

3. image Ethanol can be mixed with gasoline to reduce automobile emissions. Much automotive gasoline is 15 percent ethanol by volume. An oil company tries decreasing the ethanol content to 6 percent to lower the cost. If a car with a 14-gallon tank is filled with the 15 percent blend and a second car with a 10-gallon tank is filled with the 6 percent blend, how many times more ethanol is in the first car than in the second car?

A. 1.5

B.2.5

C.3.5

D. 4.0

4. image Kelania owns a bakery, and she adjusts the number of pounds of flour she orders each week based on the number she used the previous week. After the first week of the month, she decreased the number of pounds of flour she ordered by 25 percent, then increased the number by 10 percent the following week, then increased the number by an additional 50 percent in the last week of the month. What is the approximate total percent increase in the number of pounds of flour Kelania ordered from the start of the month until the end of the month?

A. 20%

B.24%

C.30%

D. 35%

5. image The cost of tuition at a private nonprofit four-year college in 1988 was approximately $15,800. In 2013, the cost of tuition at the same type of college was approximately $30,100. If tuition experiences the same total percent increase over the next 25 years, approximately how much will tuition at a private nonprofit four-year college cost?

A. $44,400

B.$45,800

C.$57,300

D. $66,200

Check Your Work

1. D

Difficulty: Easy

Category: Ratios and Proportions

Getting to the Answer: Assign a variable, say n, to the number of students expected to score at least 90 percent when 12,000 students take the test. Then, set up a proportion and solve for n:

image

The correct answer is (D).

2. A

Difficulty: Easy

Category: Unit Conversion

Getting to the Answer: Map out your route from starting units to end units, being mindful of the fact that the question deals with units of area (square units). The starting quantity is in square feet, and the desired quantity is in square yards. The only conversion factor you need is 3 feet (ft) = 1 yard (yd). Setting up your route to square yards, you get:

image

This matches (A).

3. C

Difficulty: Medium

Category: Percents

Getting to the Answer: Starting with the 14-­gallon tank, plug the known values into the three-part ­formula: 15% × 14 =? → 0.15 × 14 = 2.1 gallons of ethanol. Repeat for the smaller tank: 6% × 10 =? → 0.06 × 10 = 0.6 gallons of ethanol. The question asks how many times more ethanol is in the larger tank, so divide the quantities to get image. This matches (C).

4. B

Difficulty: Medium

Category: Percent Change

Getting to the Answer: Remember to avoid merely adding the percentages together. Find each change individually. You’re not given a definite number of pounds of flour in the question, so assume Kelania starts with 100. To save a step with each change, calculate the total amount instead of the weekly increase or decrease. The first change is −25%; so the number of pounds of flour ordered the next week is 75% × 100 = 0.75 × 100 = 75. The second change is +10%, which corresponds to 110% × 75 = 1.1 × 75 = 82.5 pounds. The final change is +50%, which means there are now 150% × 82.5 = 1.5 × 82.5 = 123.75 pounds. The percent change is image. Now, round to the nearest percent and you get 24%, which is (B).

5. C

Difficulty: Medium

Category: Percent Change

Getting to the Answer: Find the percent increase using the formula: image. Then, apply the same percent increase to the amount for 2013. The amount of increase is 30,100 − 15,800 = 14,300, so the percent increase is image over 25 years. If the total percent increase over the next 25 years is the same, the average cost of tuition will be 30,100 × 1.905 = 57,340.50, or about $57,300, which is (C).